Interesting "real life" applications of serious theorems

Question

As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like:

"You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $k$ persons, which means every person gets a piece of pizza with exactly $\frac{1}{k}$ of any of the $n$ topping on it."

Are there more examples which are particular interesting or instructive?

EDIT: Since this is turning into a list of mathematical jokes or sophisticated proofs for simple facts, I may have to be more precise what I was asking for: a "real life example didactically used to motivate a mathematical theorem" (thanks to Lord_Gestalter for this great wording).

Answer 1

$\sqrt[n]{2}$ is not rational for $n\geq 3$

Proof: If $\sqrt[n]{2}=\frac{p}{q}$ then $q^n+q^n=p^n$ contradicting Fermat's last theorem.

Answer 2

In a cup of coffee, one molecule of coffee is in its original location, even though the contents are undergoing convection. This is the Brouwer fixed point theorem.

Answer 3

The $n=2$ case of the Borsuk-Ulam theorem can be visualized by saying there exists some pair of antipodal points on the Earth with equal temperatures and barometric pressures. Of course, this is assuming that temperature and pressure vary continuously.

Ramsey's theorem says that, if given a sufficiently large complete graph that has been arbitrarily colored with $n$ colors, then one can find a monochromatic complete subgraph of a particular size. One example is as follows: Given any $2$-coloring on $K_6$, then we are guaranteed to find a monochromatic subgraph of size $3$. This has an interesting real-life interpretation: If we invite $6$ people to a party, then at least $3$ of them must be mutual acquaintances, or at least $3$ of them must be mutual strangers.

Answer 4

This is a fairly simple, but interesting application.

The mean-value theorem is used in some Automatic number plate recognition systems, based on average speed measurement. Using Optical character recognition, a computer system reads car plates at several fixed points along a road, and then uses the distance between these points to compute the car's average speed. By the mean-value theorem, the car must have gone at precisely that speed somewhere between the fixed points. A penalty can be issued accordingly, if this average speed exceeds the speed limit.

The system is used in several countries. It is not allowed in California, via Section 40801, Speed Trap Prohibition, of the California Vehicle Code, that reads

No peace officer or other person shall use a speed trap in arresting, or participating or assisting in the arrest of, any person for any alleged violation of this code nor shall any speed trap be used in securing evidence as to the speed of any vehicle for the purpose of an arrest or prosecution under this code.

Answer 5

I shared this one recently, and I think it is pretty funny. Using cohomology, we can prove that the alternating sum of binomial coefficients vanishes: $$\sum_{j=0}^n {n \choose j} (-1)^j=0.$$

Let $X=(S_1)^n$ be the $n$-dimensional torus. By the Künneth formula, and by the cohomology of $S^1$, $H^j(X, \mathbf Q)$ has dimension ${n \choose j}$. Therefore, the Euler characteristic of $X$ is

$$\chi(X)=\sum_{j=0}^n (-1)^j \mathrm{dim}_{\mathbf Q}H^j(X, \mathbf Q) = \sum_{j=0}^n {n \choose j} (-1)^j.$$

On the other hand, $X$ is a compact Lie group; let $\sigma$ be an infinitesimal translation $X \to X$. By the Lefschetz fixed point theorem, $\chi(X)$ is equal to the number of fixed points of $\sigma$, i.e., $0$.

Answer 6

Theorem : You can't cut a square pizza into equal-sized triangles among an odd number of persons, each of whom receive one slice.

This is a application of Sperner's lemma and the 2-adic valuation. Here is a nice proof.

Answer 7

There is the futurama theorem It is about a mind-swapping machine. The question solved is:

When is it possible to return all the minds to their original bodies?

Answer 8

I had a teacher that liked to invoke Fubini's theorem to interchange the order of two finite sums.

Answer 9

Ramsey's theorem has already been mentioned, but there is a key anecdote that deserves to be better known, and it is a great way to motivate the area. This comes from The Princeton companion to mathematics, Timothy Gowers, June Barrow-Green, and Imre Leader, eds., PUP, 2008. Chapter IV.19, Extremal and Probabilistic Combinatorics, pp. 562-575, written by Noga Alon and Michael Krivelevich, starts with the following story:

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than sociological one. Indeed, a brief discussion with the mathematicians Erdős, Turán, and Sós convinced him this was the case.

What Szalai was observing, in the language of Ramsey theory, is that $r(4,4)\le 20$, that is, any graph on $20$ vertices either contains a copy of $K_4$, or else it contains an independent set of size $4$ (actually, $r(4,4)=18$).

Alexander Bogomolny's website Cut the knot also recounts this story, and contains a proof of the equality $r(4,4)=18$, that has also been asked here a couple of times.

Another paper that recounts the story is Ramsey theory and the history of pre-Christian England, by William Gasarch. This paper goes on to present another application of this result. Unfortunately, as far as I can tell, the new story there appears to be fictional.

Answer 10

You cannot create a "good" (continuous) function that for each 3D direction (unit vector) gives you an orthogonal direction.

Seriously. You cannot construct a "good" perpendicular to a vector.

Why? Blame the Hairy ball theorem. If you could construct such function $f(\vec v) : \forall \vec v f(\vec v) \perp \vec v $, then you have essentially constructed a continuous tangent field for a sphere $\vec v \to \{\vec v \times f(\vec v)\}$ which is impossible for dimensions greater than 2.

Answer 11

Another pizza based example:

Take a slice of pizza. To keep the tip from flopping down, you pinch it between your fingers so that it gets a crease in radial direction. Works if the pizza is not too soggy.

Explanation: Theorema Egregium. Assuming the pizza is originally a rigid 2D surface with Gaussian curvature 0, it can only bend in one direction so that the product of the principal curvatures stays 0. By forcing the "harmless" curvature to be != 0, the "bad" curvature must stay 0.

Answer 12

If you get up at 7:24 AM, you miss the bus. If you get up at 6:56 AM then you have to wait too long. After many tries, you find that if you get up at 7:09 AM you arrive to the bus stop at the same time as the bus, so you sleep to the maximum but you can still get the bus. That's Bolzano's Theorem for you.

Answer 13

Olga Taussky-Todd and John Francis are two of the most recognizable figures in matrix analysis. Both were directly involved with the study of flutter (dynamic instability) of aircraft wings, while working in London. Taussky worked at the National Physical Laboratory, during World War II, and Francis worked at the National Defense Research Corporation, in the late 50s.

Studying flutter leads to delicate boundary problems in partial differential equations, but R.A. Frazer recognized its close connection with the problem of locating the eigenvalues of a matrix.

In abstract, a useful theoretical tool to attack this problem is Geršgorin's theorem, though its efficiency may vary wildly from matrix to matrix: Suppose $A$ is an $n\times n$ matrix with complex entries $a_{ij}$, $1\le i,j\le n$. If, for each $i$, we let $r_i=\sum_{j=1,j\ne i}^n|a_{ij}|$, then for each eigenvalue $\lambda$ of $A$ there is an $i$ such that $|\lambda-a_{ii}|\le r_i$. That is, the eigenvalues of $A$ are located in the union of the $n$ closed discs centered at the diagonal entries of $A$, $\{z\in\mathbb C\mid |z-a_{ii}|\le r_i\}$, $1\le i\le n$.

Taussky writes:

A large group of young girls, drafted into war work, did the calculation on hand-operated machines, following the instructions of Frazer and his assistants.

As described in section 6 of How I became a torchbearer for matrix theory, Geršgorin's theorem proved indeed to be a key tool in carrying out the relevant computations.

Once again, I didn't ask to be assigned to matrix problems. They found me.

By the time Francis came to work on the problem, the connection with computation of eigenvalues was well established, and Francis was assigned to write the relevant computer programs to carry out this task. Trying to accelerate the time the computations required, he created the shifted $QR$-algorithm which, even nowadays, is one of the most efficient tools to compute eigenvalues (and, via companion matrices, roots of arbitrary polynomials).

As David Watkins writes in Francis's algorithm:

[Francis's method] became and has continued to be the big workhorse of eigensystem computations. A version of Francis’s algorithm was used by MATLAB when I asked it to compute the eigensystem of a $1000\times 1000$ matrix on my laptop.

Answer 14

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem". Have you more examples which are particular interesting or instructive?

This may be neither interesting nor instructive, but the canonical example I have repeatedly heard of this (which was once posed to me I believe on a second year calculus final exam) is: at time zero a turkey at temperature whatever is placed some distance above a heating element at temperature whatever. Assume the heating element is an infinite flat plane and the turkey is a uniform sphere of radius whatever, and that the thermal conductivity of turkey is whatever; a spherical turkey is safe to eat when its internal temperature is everywhere greater than whatever. What is the earliest time we can safely eat the turkey?"

This is of course a variation on this joke: http://en.wikipedia.org/wiki/Spherical_cow. More on the subject can be found here: http://physics.about.com/od/physicsintherealworld/p/TurkeyPhysics.htm

In the intervening two decades I have sadly forgotten how to solve the Heat Equation. I have roasted a great many (nonspherical!) turkeys since then; I use a thermometer to tell me when its done.

I also had a professor for first year calculus who liked to use chickens to illustrate theorems. A typical moment from a lecture: "If you have an infinite plane containing an infinite number of chickens, and you have infinitely many chickens inside a pen, and finitely many chickens outside the pen (waves arms wildly to illustrate infinitely many chickens in a bounded area) you just make your arms bigger (waves even more wildly) and then you have the entire set of chickens in the pen." Which was much more memorable than "the union of an infinite but bounded set with a finite set is a bounded set."

Answer 15

I'm not sure if the various puzzles about the non-planarity of $K_{3,3}$ count as real-life. Or in fact several other things from graph theory such as the Königsberg bridge problem, or the marriage problem (where I doubt that any community "optimizes" marriage by studying a bipartite graph).

Answer 16

Euler's Theorem forms the basis of the RSA encryption system that's used in virtually every form of web-based cryptography (most notably, SSL and TLS protocols). "A Practical Application For Prime Numbers" provides an explanation better than I ever could. Regardless, the practical implications of this theorem are vast, and widely used (even right now, as I am about to submit this reply).

/I'm not a mathematician, but felt compelled to respond.

Answer 17

I'm a chemist but love maths and physics too :

Eigenvectors are a simply way to proof Descartes' laws are true using PHPW (Plane Harmonic Progressive Waves)

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The idea is to be sure that $$\left(\exp\left(\lambda_i x\right)\right)_{i \in [[0,n]]}$$

is a free familly for $\lambda_0 < ... < \lambda_n$

Which is true considering the devrivation endomorphism D for $C^\infty$ class function, then $$D(\exp(\lambda_i))=\lambda_i\times\exp(\lambda_i x)$$ we conclued saying in a $n+1$ space dimension with $n+1$ different eigenalues...

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Well now is Descartes' Laws reallife ? In physics eigenvectors are necessary too, to find solutions of Schrödinger equations, like atomic orbitals or molecular orbitals, solving a system using Hückel determinant.